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The following is an article from The Annals of Improbable Research.

by Stuart M. Shieber, Professor of Computer Science
Harvard University, Cambridge, Massachusetts, USA

I demonstrate that Hinchliffe’s rule—if the title of a scholarly article is a yes-no question, the answer is “no”—is paradoxical, by providing an article whose title is a question whose answer is “no” if and only if its answer is “yes”.

Hinchliffe’s Rule
Hinchliffe’s rule, attributed with unknown veracity to Ian Hinchliffe (Holderness, 2014), is this: “If the Title of a Scholarly Article Is a Yes-No Question, the Answer is ‘No’.” It can be seen as the academic analog of Betteridge’s law of headlines (Betteridge, 2009).

In 1988, Boris Peon (which appears, thankfully if unsurprisingly, to be a pseudonym (Strassler, 2013)) distributed an article entitled “Is Hinchliffe’s Rule True?” (Peon, 1988) Here is the abstract, which constitutes the article in its entirety:

Hinchliffe has asserted that whenever the title of a paper is a question with a yes/no answer, the answer is always no. This paper demonstrates that Hinchliffe’s assertion is false, but only if it is true.

Peon’s article hints at Hinchliffe’s rule being paradoxical, but it fails to manifest a true paradox. This is a lost opportunity, which I intend to correct by this very article.

The Problem with Peon’s Purported Paradox
The second sentence of Peon’s abstract asserts that Hinchliffe’s rule is false only if it is true, intimating a connection to the liar paradox (Beall and Glanzberg, 2014). This assertion is false.

The construction “p if q” is an alternative to “if q then p”. The converse, “ if p then q” can be presented as “p only if q”. (The conjunction of the two, “p if and only if q”, therefore expresses the biconditional.) Peon’s postulation can thus be reconstructed as stating “if Hinchliffe’s rule is false then it is true”.

But this does not follow from the existence of Peon’s article. If Hinchliffe’s rule is false, we can deduce nothing about the truth or falsity of that article’s title, that is to say, nothing about the truth or falsity of Hinchliffe’s rule, and certainly not that the rule is true. Only the other direction of reasoning holds, that if Hinchliffe’s rule is true, then it is false: Suppose Hinchliffe’s rule is true. Then the answer to the article’s title ought to be “yes”. But by Hinchliffe’s rule, the answer to the article’s title must be “no”. The contradiction leads us to conclude that Hinchliffe’s rule must be false, a conclusion that is self-consistent. (Of course, the same conclusion would follow from any article title in the form of a question with a positive answer, whether it mentions Hinchliffe’s rule or not.) No paradox is exemplified by Peon’s article.1

A Lost Opportunity Regained
Peon purports to demonstrate Hinchliffe’s rule’s paradoxical nature, but unfortunately has failed. The path to a paradox lies in the distinction between what the rule requires and what it is consistent with. Consider an article with a title question phrased as the negation of Peon’s, viz., “Is Hinchliffe’s Rule False?” If Hinchliffe’s rule is true, then the answer to the title question is “no”, which is consistent with the assumption, although it does not require it. If Hinchliffe’s rule is false, then the answer to the title question is “yes”, which is again consistent with the assumption, although it does not require it. We can conclude from such an article nothing about the truth of Hinchliffe’s rule, but this case at least exhibits symmetry between the two horns of the dilemma, which will be needed in a paradoxical application of Hinchliffe’s rule. Consistency with Hinchliffe’s rule is the foundation on which to erect a paradox.

(Image credit: Flickr user Visionello)

Taking a clue from this observation, consider a final example, an article—such as this very one—with the title “Is This Article Consistent with Hinchliffe’s Rule?” Suppose the article is consistent with Hinchliffe’s rule. Then the answer to the title question ought to be “no”, that is, the article is not consistent with Hinchliffe’s rule. Conversely, suppose the article is not consistent with Hinchliffe’s rule. Then the answer to the title question must not be “no” (in other words, must be “yes”), stating that the article is consistent with Hinchliffe’s rule. In either case, we have a contradiction, and Hinchliffe’s rule leads to a paradox.

Note
1. Perhaps the conditional in Peon’s abstract was intended to be interpreted as encompassing its converse as well via so-called “conditional perfection” (Horn, 2000), the kind of reasoning that allows moving from “I’ll give you five bucks if you shut up” to the implicated condition “I’ll give you five bucks only if you shut up”. But this strengthened interpretation as a biconditional (“Hinchliffe’s rule is false if and only if it is true”) is also not paradoxical, merely false.

References
Beall, Jc, and Michael Glanzberg (2014). “Liar Paradox.” In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, Fall 2014.

Betteridge, Ian. 2009. “TechCrunch: Irresponsible Journalism.” Technovia.

Holderness, Mike (2014). “Feedback.” New Scientist, vol. 223 (2982) (16 August): 48.

Horn, Laurence R. (2000). “From if to iff: Conditional Perfection as Pragmatic Strengthening.” Journal of Pragmatics, 32 (3): 289–326.

Peon, Boris (1988). “Is Hinchliffe’s Rule True?” Manuscript.

Strassler, Matt (2013). “A Second Higgs Particle?” Of Particular Significance.

_____________________

The article above is from the May-June 2015 issue of the Annals of Improbable Research. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!

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